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On a compact domain $\Omega \subseteq \mathbb{R}^d$, we have a function $u(x) \in C^{\infty}(\Omega)$ with an approximation $u_h(x)$ with the following properties:

$$ |u(x) - u_h (x)| \leq C h^{m+1} |u(x)|_{C^{m+1}(\Omega)}, $$

where the seminorm on the right is defined as $ |u(x)|_{C^{m+1}(\Omega)} = \text{max}_{|\alpha|=m+1} \|D^{\alpha} u\|_{L^{\infty}} $, with the derivative being defined using the multi-index notation:

$$ D^{\alpha} f = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \cdots\partial x_n^{\alpha_n}}, \quad \alpha = (\alpha_1, \cdots, \alpha_n), $$

for $|\alpha| = \alpha_1 + \alpha_2 + \cdots \alpha_n| \leq k$, $x \in \mathbb{R}^d$, $f \in C^k (\Omega)$. I want to use that approximation to derive similar bound for the seminorm $|v|^2_{k, \Omega}$ defined as the sum of all the squares of the $L^2$ norms of all the derivatives of order $k$. So in 2D, this seminorm would look like: $$ \|v\|^2_{0, \Omega} = |v|^2_{0, \Omega} = \int_{\Omega} v^2 d\Omega \\ |v|^2_{1, \Omega} = \left| \frac{\partial v}{\partial x_1} \right|^2_{0, \Omega} + \left| \frac{\partial v}{\partial x_2} \right|^2_{0, \Omega} \\ |v|^2_{2, \Omega} = \left| \frac{\partial^2 v}{\partial x_1^2} \right|^2_{0, \Omega} + \left| \frac{\partial^2 v}{\partial x_1 \partial x_2} \right|^2_{0, \Omega} + \left| \frac{\partial^2 v}{\partial^2 x_2} \right|^2_{0, \Omega} $$ To be more precise, I'm trying to arrive at some bound for $|u(x) - u_h(x)|_{s, \Omega}$, with $0 \leq s < m$. Any suggestions how I could achieve that?

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  • $\begingroup$ Hi from Vietnam $\endgroup$
    – haidangel
    Mar 22 at 6:46
  • $\begingroup$ Use a suitable bump function. That is pick $\phi\in C^\infty(\Omega)$ such that $\phi=1$ on the interior of $\Omega$ and $\phi=0$ on $\partial\Omega$. Then use repeated applications of integration by parts to put all the derivatives on $\phi$ and then use the assumed bound on $|u-u_h(x)|$. $\endgroup$ Mar 30 at 16:23

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